Alphabetical free group elements
alpha.RdProduces simple vectors of free group elements based on the alphabet
Details
Function alpha() takes an integer i and returns the
letter i of the alphabet. Thus alpha(3) returns
c. The function is vectorised: alpha(1:3) returns
a b c.
Function abc() takes an integer i and returns letters
1 to i of the alphabet. Thus abc(4) returns
a.b.c.d. The function is vectorised.
Remember that “letters of the alphabet” is just a phrase: above it refers to the default print method which can be changed, see the examples.
Examples
alpha(5) # just the single letter 'e'
#> [1] e
abc(5) # product of a,b,c,d,e
#> [1] a.b.c.d.e
alpha(1:26) # the whole alphabet; c
#> [1] a b c d e f g h i j k l m n o p q r s t u v w x y z
all(alpha(1:26) == as.free(letters)) # should be TRUE
#> [1] TRUE
z <- alpha(26) # variable 'z' is symbol 26, aka 'z'.
abc(1:10) ^ z
#> [1] z^-1.a.z z^-1.a.b.z
#> [3] z^-1.a.b.c.z z^-1.a.b.c.d.z
#> [5] z^-1.a.b.c.d.e.z z^-1.a.b.c.d.e.f.z
#> [7] z^-1.a.b.c.d.e.f.g.z z^-1.a.b.c.d.e.f.g.h.z
#> [9] z^-1.a.b.c.d.e.f.g.h.i.z z^-1.a.b.c.d.e.f.g.h.i.j.z
abc(-5:5)
#> [1] e^-1.d^-1.c^-1.b^-1.a^-1 d^-1.c^-1.b^-1.a^-1 c^-1.b^-1.a^-1
#> [4] b^-1.a^-1 a^-1 0
#> [7] a a.b a.b.c
#> [10] a.b.c.d a.b.c.d.e
alpha(-5:5)
#> [1] e^-1 d^-1 c^-1 b^-1 a^-1 0 a b c d e
sum(abc(-5:5))
#> [1] 0
## bear in mind that the symbols used are purely for the print method:
jj <- LETTERS[1:10]
options(freegroup_symbols = apply(expand.grid(jj,jj),1,paste,collapse=""))
alpha(c(66,67,68,69)) # sensible output
#> [1] FG GG HG IG
options(freegroup_symbols=NULL) # restore to symbols to default letters
alpha(c(66,67,68,69)) # print method not very helpful now
#> [1] NA NA NA NA